Andreas Eberle

Absence of spectral gaps on a class of loop spaces

Abstract Let M be a compact simply connected Riemannian manifold which contains a non-trivial closed geodesic γ such that the curvature is constant and strictly negative in a neighbourhood of γ . We show that in this case a Poincare inequality for the H 1 gradient on the free loop space C ( S 1 , M ) endowed with Bismut measure does not hold. Similar results hold w.r.t. more general metrics, and also on based loop spaces with base point close to γ . A key ingredient in the proofs is a result which shows that in a certain sense, the concentration of a Brownian bridge on hyperbolic space near the geodesic joining the end points x and y increases rapidly as d ( x , y )→∞.

Reflection couplings and contraction rates for diffusions

We consider contractivity for diffusion semigroups w.r.t. Kantorovich (

Reflection coupling and Wasserstein contractivity without convexity

L^1

Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions

L1 Wasserstein) distances based on appropriately chosen concave functions. These distances are inbetween total variation and usual Wasserstein distances. It is shown that by appropriate explicit choices of the underlying distance, contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w.r.t. standard Wasserstein distances fails. Applications include overdamped Langevin diffusions with locally non-convex potentials, products of these processes, and systems of weakly interacting diffusions, both of mean-field and nearest neighbour type.